In this paper, the hereditarily almost expandability of inverse limits is investigated, with two results obtained. Let X be the inverse limit space of an inverse system (Xα, π^αβ, ∧}. (1) Suppose X is heredita...In this paper, the hereditarily almost expandability of inverse limits is investigated, with two results obtained. Let X be the inverse limit space of an inverse system (Xα, π^αβ, ∧}. (1) Suppose X is hereditarily κ-metacompact, if each Xα is hereditarily pointwise collectionwise normal (almost θ-expandable, almost discrete θ-expandable), then so is X; (2) Suppose X is hereditarily κ-σ-metacompact, if each Xα is hereditarily almost σ-expandable (almost discrete σ-expandable = σ-pointwise collectionwise normal), then so is X.展开更多
Let X be the limit of an inverse system {Xα, παβ, ∧} and and let λ be the cardinal number of A. Assume that each projection πα : X → Xα is an open and onto map and X is A-paracompact. We prove that if each ...Let X be the limit of an inverse system {Xα, παβ, ∧} and and let λ be the cardinal number of A. Assume that each projection πα : X → Xα is an open and onto map and X is A-paracompact. We prove that if each Xα is B(LF, ω^2)-refinable (hereditarily B(LF, ω^2)- refinable), then X is B(LF, ω^2)-refinable (hereditarily B(LF,ω ^2)-refinable). Furthermore, we show that B(LF, ω^2)-refinable spaces can be preserved inversely undcr closed maps.展开更多
This paper proves the following results: Le t X= lim ←{X σ,π σ ρ,Λ},|Λ|=λ, and every p rojection π σ: X→X σ be an open and onto mapping. (A) If X is λ-paracompact and every X σ is normal and δθ-ref...This paper proves the following results: Le t X= lim ←{X σ,π σ ρ,Λ},|Λ|=λ, and every p rojection π σ: X→X σ be an open and onto mapping. (A) If X is λ-paracompact and every X σ is normal and δθ-refinable, then X is normal and δθ-refinable; (B) If X is hereditarily λ-pa racompact and every X σ is hereditarily normal and hereditarily δθ- refinable, then X is hereditarily normal and hereditarily δθ-refiable .展开更多
Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p 〉 O. Then the inverse limits of "higher" reduced enveloping algebras {uxs (g) I s ∈ N} with X running over g* ma...Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p 〉 O. Then the inverse limits of "higher" reduced enveloping algebras {uxs (g) I s ∈ N} with X running over g* make representations of g split into different "blocks". In this paper, we study such an infinite- dimensional algebra Ax (g) :=lim Uxs (g) for a given X C g*. A module category equivalence is built between subcategories of U(g)-rnod and Ax(g)-mod. In the case of reductive Lie algebras, (quasi) generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized x-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity.展开更多
The connection between APOTP (asymptotic pseudo orbit tracing property) for a continuous map on a compact metric space and that for the shift map on the inverse limit space is investigated. As an application, it is ...The connection between APOTP (asymptotic pseudo orbit tracing property) for a continuous map on a compact metric space and that for the shift map on the inverse limit space is investigated. As an application, it is showed that the shift map on Henderson pseudoarc has APOTP.展开更多
In this paper twe prove that the inverse limit of metra-projective modules (meta-injective modules resp. ) is also meta-projective (meta-injective resp. ). Let K be a field f R1, R2 be K-algebras, we also obtain a suf...In this paper twe prove that the inverse limit of metra-projective modules (meta-injective modules resp. ) is also meta-projective (meta-injective resp. ). Let K be a field f R1, R2 be K-algebras, we also obtain a sufficient condition for lgldim (R1 R2,)≥lgldim R1+lgldimR2, and wgldim (R1 R2) ≥wgldimR1 +wgldimR2展开更多
Given a compact Hausdorff space X, U(X) denotes the compact Hausdorff space of all the upper semicontinuous functions from X to the unit interval with the dual lim inf topology. Then U is an endofunctor o...Given a compact Hausdorff space X, U(X) denotes the compact Hausdorff space of all the upper semicontinuous functions from X to the unit interval with the dual lim inf topology. Then U is an endofunctor on compact Hausdorff space. It is proved in this note that this functor preserves inverse limits.展开更多
Barge asked if each homeomorphism of a hereditarily decomposable chainable continuum has zero topological entropy and Toledo asked if a homeomorphism of a chainable continuum can always be induced by square commuting ...Barge asked if each homeomorphism of a hereditarily decomposable chainable continuum has zero topological entropy and Toledo asked if a homeomorphism of a chainable continuum can always be induced by square commuting diagram on inverse systems of finite graphs. We show in this note that if Toledos question has a positive answer then Barges question also has a positive answer.展开更多
Let {Xi,πki,ω} be an inverse sequence and X -- lim{Xi,πki,ω). If each Xi is hereditarily (resp. metaLindelSf, σ-metaLindelSf, σ-orthocompact, weakly suborthocompact, δθ-refinable, weakly θ-refinable, weakly...Let {Xi,πki,ω} be an inverse sequence and X -- lim{Xi,πki,ω). If each Xi is hereditarily (resp. metaLindelSf, σ-metaLindelSf, σ-orthocompact, weakly suborthocompact, δθ-refinable, weakly θ-refinable, weakly δθ-refinable), then so is X.展开更多
A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of count...A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.展开更多
Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_...Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_1=max(A\{a}),and B={x∈[c_2,1]|f(x)=x},b=minB,b_1=min(B\{b}).Then the in- verse limit(I,f)is an arc if and only if one of the following three conditions holds: (1)If c_1<f(c_1)≤c_2(resp.c_1≤f(c_2)<c_2),then f has a single fixed point,a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods,furthermore,if A≠φ and B≠φ,then f(c2)>a(resp.f(c_1)<b). (2)If f(c_1)≤c_1(resp.f(c_2)≥c_2),then f has more than one fixed point,furthermore,if B≠φ and A\{a}≠φ,f(c_2)≥a or if a_1<f(c_2)<a,f^2(c_2)>f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if b<f(c_2)<b_1,f^2(c_1)<f(c_1)). (3)If f(c_1)>c_2 and f(c_2)<c_1,then f has a single fixed point,a single period two orbit lying in I\(u,v)but no points of period greater than two,where u,v ∈[c_1,c_2] such that f(u)=c_2 and f(v)=c_1.展开更多
Let G be a graph which contains exactly one simple closed curve.We prove that a continuous map f:G→G has zero topological entropy if and only if there exist at most k■[(Edg(G)+End(G)+ 3)/2]different odd numbers n_1,...Let G be a graph which contains exactly one simple closed curve.We prove that a continuous map f:G→G has zero topological entropy if and only if there exist at most k■[(Edg(G)+End(G)+ 3)/2]different odd numbers n_1,...,n_k such that Per(f)is contained in ∪_i^k=1 ∪_j~∞=0 n_i2~j,where Edg(G) is the number of edges of G and End(G)is the number of end points of G.展开更多
基金Supported by the Scientific Fund of the Educational Committee of Xinjiang of China (XJEDU2004158)
文摘In this paper, the hereditarily almost expandability of inverse limits is investigated, with two results obtained. Let X be the inverse limit space of an inverse system (Xα, π^αβ, ∧}. (1) Suppose X is hereditarily κ-metacompact, if each Xα is hereditarily pointwise collectionwise normal (almost θ-expandable, almost discrete θ-expandable), then so is X; (2) Suppose X is hereditarily κ-σ-metacompact, if each Xα is hereditarily almost σ-expandable (almost discrete σ-expandable = σ-pointwise collectionwise normal), then so is X.
基金Supported by the National Natural Science Foundation of China (10671173)
文摘Let X be the limit of an inverse system {Xα, παβ, ∧} and and let λ be the cardinal number of A. Assume that each projection πα : X → Xα is an open and onto map and X is A-paracompact. We prove that if each Xα is B(LF, ω^2)-refinable (hereditarily B(LF, ω^2)- refinable), then X is B(LF, ω^2)-refinable (hereditarily B(LF,ω ^2)-refinable). Furthermore, we show that B(LF, ω^2)-refinable spaces can be preserved inversely undcr closed maps.
文摘This paper proves the following results: Le t X= lim ←{X σ,π σ ρ,Λ},|Λ|=λ, and every p rojection π σ: X→X σ be an open and onto mapping. (A) If X is λ-paracompact and every X σ is normal and δθ-refinable, then X is normal and δθ-refinable; (B) If X is hereditarily λ-pa racompact and every X σ is hereditarily normal and hereditarily δθ- refinable, then X is hereditarily normal and hereditarily δθ-refiable .
基金Supported by National Natural Science Foundation of China (Grant Nos. 11126062,11201293 and 11271130)the Innovation Program of Shanghai Municipal Education Commission (Grant Nos. 12ZZ038 and 13YZ077)
文摘Let (g, [p]) be a restricted Lie algebra over an algebraically closed field of characteristic p 〉 O. Then the inverse limits of "higher" reduced enveloping algebras {uxs (g) I s ∈ N} with X running over g* make representations of g split into different "blocks". In this paper, we study such an infinite- dimensional algebra Ax (g) :=lim Uxs (g) for a given X C g*. A module category equivalence is built between subcategories of U(g)-rnod and Ax(g)-mod. In the case of reductive Lie algebras, (quasi) generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized x-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity.
基金Supported by the NNSF of China (10361001) the NSF of the Education Committee of Jiangsu Province (05KJB110033)Young Teachers' Project of College of Anhui Province (2006jqll62).
文摘The connection between APOTP (asymptotic pseudo orbit tracing property) for a continuous map on a compact metric space and that for the shift map on the inverse limit space is investigated. As an application, it is showed that the shift map on Henderson pseudoarc has APOTP.
文摘In this paper twe prove that the inverse limit of metra-projective modules (meta-injective modules resp. ) is also meta-projective (meta-injective resp. ). Let K be a field f R1, R2 be K-algebras, we also obtain a sufficient condition for lgldim (R1 R2,)≥lgldim R1+lgldimR2, and wgldim (R1 R2) ≥wgldimR1 +wgldimR2
文摘Given a compact Hausdorff space X, U(X) denotes the compact Hausdorff space of all the upper semicontinuous functions from X to the unit interval with the dual lim inf topology. Then U is an endofunctor on compact Hausdorff space. It is proved in this note that this functor preserves inverse limits.
文摘Barge asked if each homeomorphism of a hereditarily decomposable chainable continuum has zero topological entropy and Toledo asked if a homeomorphism of a chainable continuum can always be induced by square commuting diagram on inverse systems of finite graphs. We show in this note that if Toledos question has a positive answer then Barges question also has a positive answer.
文摘Let {Xi,πki,ω} be an inverse sequence and X -- lim{Xi,πki,ω). If each Xi is hereditarily (resp. metaLindelSf, σ-metaLindelSf, σ-orthocompact, weakly suborthocompact, δθ-refinable, weakly θ-refinable, weakly δθ-refinable), then so is X.
文摘A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.
基金Supported by the National Natural Science Foundation of China(No.19961001,No.60334020)Outstanding Young Scientist Research Fund.(No.60125310)
文摘Let I=[0,1],c_1,c_2 ∈(0,1)with c_1<c_2 and f:I→I be a continuous map satisfying:f|_[0,c_1] and f|_[c_2,1] are both strictly increasing and f|_[c_1,c_2]is strictly decreasing.Let A ={z ∈[0,c_1]|f(x)=x}, a=maxA,a_1=max(A\{a}),and B={x∈[c_2,1]|f(x)=x},b=minB,b_1=min(B\{b}).Then the in- verse limit(I,f)is an arc if and only if one of the following three conditions holds: (1)If c_1<f(c_1)≤c_2(resp.c_1≤f(c_2)<c_2),then f has a single fixed point,a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods,furthermore,if A≠φ and B≠φ,then f(c2)>a(resp.f(c_1)<b). (2)If f(c_1)≤c_1(resp.f(c_2)≥c_2),then f has more than one fixed point,furthermore,if B≠φ and A\{a}≠φ,f(c_2)≥a or if a_1<f(c_2)<a,f^2(c_2)>f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if b<f(c_2)<b_1,f^2(c_1)<f(c_1)). (3)If f(c_1)>c_2 and f(c_2)<c_1,then f has a single fixed point,a single period two orbit lying in I\(u,v)but no points of period greater than two,where u,v ∈[c_1,c_2] such that f(u)=c_2 and f(v)=c_1.
基金Project supported by NSF (10171034) of ChinaNSF (970395) of Guangdong province
文摘Let G be a graph which contains exactly one simple closed curve.We prove that a continuous map f:G→G has zero topological entropy if and only if there exist at most k■[(Edg(G)+End(G)+ 3)/2]different odd numbers n_1,...,n_k such that Per(f)is contained in ∪_i^k=1 ∪_j~∞=0 n_i2~j,where Edg(G) is the number of edges of G and End(G)is the number of end points of G.